Compressed Sensing and the Universe's Sparse Basis Functions: Fourier, Gaussian, Hermite, Sech, and Airy

Forum topic · 小凯 · 2026-08-26

Summary

This Chinese tech forum post explores why the physical world can be compressed into sparse representations through the lens of compressed sensing. The author argues that complex continuous physical fields exhibit extreme sparsity on specific self-dual basis functions: Fourier plane waves, Gaussians, Hermite-Gaussian modes, hyperbolic sec (sech) solitons, and Airy functions. It explains the compressed sensing framework, where a K-sparse signal can be recovered from M ≈ O(K log(N/K)) incoherent measurements via convex optimization. Fourier and Gaussian bases are shown to derive from a common generalized quadratic exponential family, with Fourier serving as eigenfunctions of the derivative operator and Gaussians achieving the Heisenberg uncertainty lower bound with strict Fourier self-duality. These theoretical insights are tied to two modern engineering breakthroughs: the Fourier Neural Operator (FNO) for fast PDE solving via low-frequency mode truncation, and 3D Gaussian Splatting (3DGS) for real-time rendering using sparse surface-based Gaussian primitives. References include foundational papers by Candès, Donoho, Li et al. (ICLR 2021), and Kerbl et al. (SIGGRAPH 2023).

Overview

Imagine the universe as a giant projector, with infinite-dimensional electromagnetic vibrations, quantum transitions, and fluid waves emerging every second. Recording all of it point-by-point would exceed any storage capacity. Yet the physical world exhibits astonishing simplicity when expanded on specific basis functions.

Compressed sensing reveals this secret: natural laws are knowable because complex physical fields are extremely sparse on Fourier functions, Gaussians, and self-dual function families such as Hermite and hyperbolic sec functions. This space-frequency self-dual symmetry allows full reconstruction of continuous worlds from very few incoherent measurements via convex optimization.

1. The Core Law of Compressed Sensing

Classical information theory requires sampling at twice the highest frequency (the Nyquist limit). Compressed sensing breaks this barrier: if a signal has only K significant nonzero components (K-sparse) under an orthogonal basis Ψ, then with an incoherent measurement matrix Φ, only M ≈ 𝒪(K log(N/K)) samples are needed for accurate recovery:

\min_{\boldsymbol{\theta}} \|\boldsymbol{\theta}\|_1 \quad \text{s.t.} \quad \mathbf{y} = \mathbf{\Phi} \mathbf{\Psi} \boldsymbol{\theta}

Key concepts:

Sparsity basis Ψ: an orthogonal function system whose expansion coefficients are mostly zero or rapidly decaying.
Incoherence μ: the maximum inner product between rows of Φ and columns of Ψ; the lower the coherence, the more global sparse information each measurement carries.

The sparse basis family includes: Fourier complex exponentials (eigenbasis of periodic waves and differential operators, core of FNO), anisotropic Gaussians (space-frequency compactness and geometric surfaces, core of 3DGS), Hermite-Gaussian functions (exact Fourier transform eigenbasis, quantum harmonic oscillator modes), and hyperbolic sec (sech) functions (nonlinear dispersion-canceling self-dual basis, optical soliton communication).

2. Fourier and Gaussian: Space-Frequency Duality of the Generalized Quadratic Exponential Family

Fourier and Gaussian functions originate from the same generalized quadratic exponential family:

f(x) = \exp\left( -a x^2 + b x + c \right), \quad a, b, c \in \mathbb{C}, \quad \text{Re}(a) \ge 0

With a = 0, b = ik: Fourier waves e^(ikx) — narrow in frequency, oscillating globally in space (global wave sparsity).
With a > 0, b = 0: real Gaussian e^(-ax²) — balanced convergence in both domains, achieving the Heisenberg lower bound (local geometric sparsity).

3. Self-Dual Function Families

Hermite-Gaussian functions ψ_n(x) = H_n(x) e^(−x²/2) form the complete orthogonal eigenfunction system of the Fourier transform in Hilbert space:

\mathcal{F}\{ H_n(x) e^{-x^2 / 2} \} = (-i)^n H_n(\omega) e^{-\omega^2 / 2}

Quantum harmonic oscillator energy levels correspond exactly to Hermite-Gaussian modes; laser cavity TEM_mn transverse modes are sparse linear combinations of them.

Hyperbolic secant (sech) has the same Fourier self-duality as the Gaussian:

\mathcal{F}\{ \text{sech}(\pi x) \} = \text{sech}(\pi \omega)

When group velocity dispersion and self-phase modulation reach dynamic balance in nonlinear optical fiber, pulses lock into sech soliton shapes that travel thousands of kilometers undistorted.

Airy function Ai(x), from the cubic-phase integral Ai(x) = (1/π)∫₀^∞ cos(t³/3 + xt) dt, forms self-bending, diffraction-free beams that curve along parabolic trajectories and self-heal after encountering obstacles.

4. Two Engineering Breakthroughs: FNO and 3DGS

1. Fourier Neural Operator (FNO): Navier-Stokes energy spectra decay exponentially with frequency. FNO truncates high frequencies, keeping only the first k_max low-frequency modes, converting spatial integrals into frequency-domain products — thousands of times faster than traditional solvers. 2. 3D Gaussian Splatting (3DGS): matter in 3D scenes exists only on surfaces. 3DGS replaces trillions of empty voxels with millions of compact Gaussian ellipsoids, achieving closed-form projection via the Fourier slice theorem and 100+ FPS real-time rendering.

5. Conclusion

These functions recur throughout science and engineering because they touch the underlying symmetries of the physical world. Compressed sensing shows that choosing self-dual bases isomorphic to physical laws compresses seemingly infinite information complexity into minimal sparse representations — a cornerstone of AI for Science and next-generation physical vision computing.

References

1. Candès, E. J., & Wakin, M. B. (2008). *An introduction to compressive sampling*. IEEE Signal Processing Magazine, 25(2), 21-30. 2. Donoho, D. L. (2006). *Compressed sensing*. IEEE Transactions on Information Theory, 52(4), 1289-1306. 3. Li, Z., et al. (2021). *Fourier Neural Operator for Parametric Partial Differential Equations*. ICLR 2021 (Oral). 4. Kerbl, B., et al. (2023). *3D Gaussian Splatting for Real-Time Radiance Field Rendering*. ACM TOG (SIGGRAPH 2023). 5. Dym, H., & McKean, H. P. (1972). *Fourier Series and Integrals*. Academic Press.